THE INTERACTION OF SHOCKS WITH DISPERSIVE WAVES - 2 - INCOMPRESSIBLE-INTEGRABLE LIMIT

This is the second in a two-part series of articles in which we analyz e a system similar in structure to the well-known Zakharov equations f rom weak plasma turbulence theory, but with a nonlinear conservation e quation allowing finite time shock formation. In this article we analy ze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact so lutions of the full system, a multiscale perturbation method is used t o resolve several basic wave interactions. The analysis breaks down in to two categories: the nonlinear limit and the linear limit, correspon ding to the form of the equations when the group velocity to shock spe ed ratio, denoted by epsilon, is zero. The former case is an integrabl e limit in which the model reduces to the cubic nonlinear Schrodinger equation governing the dispersive wave envelope. We focus on the inter action of a ''fast'' shock wave and a single hump soliton. In the latt er case, the epsilon = 0 problem reduces to the linear Schrodinger equ ation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactio ns at lowest orders, we first analyze a simpler system of ordinary dif ferential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and deve lop a multiscale asymptotic method to derive the effective leading-ord er shock equations and the leading-order modulation equations governin g the phase and amplitude of the dispersive wave envelope. The leading -order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shoc k passes through the soliton, (b) the shock is completely blocked by t he soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the for mation of a second, transient shock front in the rapidly moving hyperb olic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manif ested, to leading order, in the generalized frequency of the dispersiv e wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer. I n the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performe d on the leading-order shock equations.